CHARACTERIZATION OF THE GROUPS D_p+1(2) AND D_p+1(3) USING ORDER COMPONENTS

Abstract

Abstract. In this paper we will prove that the groups D p+1 (2) and D p+1 (3), where p is an odd prime number, are uniquely determined by their sets of order components. A main consequence of our result is the validity of Thompson's conjecture for the groups D p+1 (2) and D p+1 (3).

On Thompson’s conjecture for finite simple groups

“…From Lemma 16, it follows that |S| < |S| 3 t . It follows from Lemmas 22,24 and 27 that |G| is divisible by (k 1 (q)k 2 (q)) 4 (k 4 (q)) 2 k 3 (q)k 6 (q). Lemma 17 implies that t ∈ R 3 (q) ∪ R 6 (q).…”

“…The prime graph 2 B 2 has 4 connected components and recognition of that's groups was proved in [20]. In [24], the recognizability of the groups D p+1 (2) and D p+1 (3) is proved, where p is a prime. In [4], the Thompson's conjecture was proved for groups of type D n (q) where n ∈ {4, 8}.…”